A State-Space Approach to the Small Gain and Passivity Theorems

Wieringa, Lars (2025) A State-Space Approach to the Small Gain and Passivity Theorems. Bachelor's Thesis, Applied Mathematics.

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Abstract

The small gain theorem is a classical result in robust control which can be used to guarantee that the output feedback interconnection of two linear time-invariant systems is well-posed and internally stable. It also has an interesting converse, which ensures boundedness of the $\mathcal H_\infty$-norm of an internally stable system under the condition that its interconnection with all other internally systems satisfying a certain $\mathcal H_\infty$-norm bound is well-posed and internally stable. The most well-known proof of this result is due to K. Zhou and J. C. Doyle, and relies on an input-output framework (i.e., transfer matrices). In this thesis, we reformulate the small gain theorem using a state-space approach and provide a novel proof of this result using the theorems of alternatives proven by V. Balakrishnan and L. Vandenberghe and the bounded real lemma. Finally, we attempt to generalize this approach and present some partial results that work towards a state-space passivity theorem.

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Waarde, H.J. van and Camlibel, M.K.
Degree programme:	Applied Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	04 Jul 2025 06:49
Last Modified:	04 Jul 2025 06:49
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/35782

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